Qinghe Yin Australian National University
βtransformations, iterated function systems, and Cantortype sets βtransformations and βexpansions were
first introduced by Renyi (1957) and further explored by Parry (1961). Given β > 1, the βtransformation
T_{β} : [0,1) → [0,1) is defined by T_{β}x = βx (mod 1). It possesses an absolutely continuous invariant measure and
is ergodic. The βexpansion of x [0,1) is determined by T_{β}. This introduces a symbolic dynamical system Σ_{β}
with onesided shift map. An iterated function system (IFS) is an (n + 1)tuple (X;f_{0},…,f_{n1}), where X is a
compact metric space and each f_{i} is a contractive map. The theory of IFS’s was first explored by
Hutchinson (1981). Many important fractals such as the Sierpinski triangle and the Cantor middlethird
set appear as attractors of certain IFS’s. We define the βattractor E_{β} for an IFS as a compact
subset, determined by Σ_{β}, of the attractor of the IFS. In the case that all f_{i} are similitudes with
contractive ratio 0 < r < 1 and with a disjoint condition, the Hausdorff dimension of E_{β} is given
by
As an application of this result, we compute the Hausdorff dimension for Cantortype sets constructed from
βexpansions with β > 2. This is a joint work with John Hutchinson.
